Date of Award

Summer 2025

Document Type

Open Access Dissertation

Department

Mathematics

First Advisor

Linyuan Lu

Abstract

This dissertation explores a variety of extremal problems in spectral graph theory, hypergraph Tur\'an theory in the $\ell_2$-norm, and Lin-Lu-Yau Ricci curvature. We begin by studying the maximization of linear combinations of eigenvalues in relation to the adjacency matrix of a graph. For a graph $G$ on $n$ vertices, we order its eigenvalues $\lambda_1 \geq \dots \geq \lambda_n$. First, we study a generalization of the maximum spread. Specifically, we consider the $(i,j)$-spread, defined as $\lambda_{i+1}-\lambda_{n-j}$, and aim to maximize that quantity over all simple graphs on $n$ vertices. We establish asymptotic bounds for all $i,j\geq 0$ and identify infinitely many pairs for which the bounds are tight. Our key insight is to focus on graphs with possible self loops. Defining $\lambda_{k,max}(n)$ as the maximum $k$th largest eigenvalue over all connected outerplanar graphs on $n$ vertices, we show that for $n$ sufficiently large, any extremal outerplanar graph achieving $\lambda_{k,max}(n)$ has exactly $k$ vertices of degree $\frac{n}{k} \pm O(1)$ and $ \lambda_{k,\max}(n) = \sqrt{n/k} + 1 + O(1/\sqrt{n}).$ Furthermore, we determine the extremal graphs achieving $\lambda_{2,max}(n)$ and establish the maximum $\lambda_2$ among all 2-connected outerplanar graphs on $n$ vertices. We shift our focus to the study of hypergraph Tur\'an problems through the lens of the $\ell_2$-norm of the codegree vector. We prove an $\ell_2$-norm version of the Erd\H{o}s-Ko-Rado theorem for all $n\geq 2k$ and establish $\ell_2$-norm versions of the Erd\H{o}s Matching Conjecture and the $t$-intersection Erd\H{o}s-Ko-Rado theorem when $n$ is sufficiently large. We determine the exact codegree squared extremal number for minimal and linear 3-paths and 3-cycles, and provide asymptotic results for minimal and linear $s$-paths and $s$-cycles when $s\geq 4$. Our main tool is an inequality of Bey, which gives a general upper bound on the codegree squared density. We also derive several exact and asymptotic results for graph Tur\'an-type problems using known spectral extremal results and Hofmeister's inequality. Lastly, we consider outerplanar graphs with positive Lin-Lu-Yau (LLY) Ricci curvature. We prove that if $G$ is a positively curved outerplanar graph with $\delta(G) \geq 2$, then $\Delta(G) \leq 9$. Additionally, for positively curved maximal outerplanar graphs with $\delta(G) \geq 2$, we have $|G|\leq 10$. Both bounds are achieved by the fan graph on 10 vertices.

Rights

© 2025, George Henry Brooks

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